In this work, we extend the data-driven It\^{o} stochastic differential equation (SDE) framework for the pathwise assessment of short-term forecast errors to account for the time-dependent upper bound that naturally constrains the observable historical data and forecast. We propose a new nonlinear and time-inhomogeneous SDE model with a Jacobi-type diffusion term for the phenomenon of interest, simultaneously driven by the forecast and the constraining upper bound. We rigorously demonstrate the existence and uniqueness of a strong solution to the SDE model by imposing a condition for the time-varying mean-reversion parameter appearing in the drift term. The normalized forecast function is thresholded to keep such mean-reversion parameters bounded. The SDE model parameter calibration also covers the thresholding parameter of the normalized forecast by applying a novel iterative two-stage optimization procedure to user-selected approximations of the likelihood function. Another novel contribution is estimating the transition density of the forecast error process, not known analytically in a closed form, through a tailored kernel smoothing technique with the control variate method. We fit the model to the 2019 photovoltaic (PV) solar power daily production and forecast data in Uruguay, computing the daily maximum solar PV production estimation. Two statistical versions of the constrained SDE model are fit, with the beta and truncated normal distributions as proxies for the transition density. Empirical results include simulations of the normalized solar PV power production and pathwise confidence bands generated through an indirect inference method. An objective comparison of optimal parametric points associated with the two selected statistical approximations is provided by applying the innovative kernel density estimation technique of the transition function of the forecast error process.

翻译：本文拓展了数据驱动伊藤随机微分方程（SDE）框架，用于短期预测误差的路径评估，以考虑自然约束观测历史数据与预测的时间相关上界。针对该现象提出一种新型非线性、非时齐SDE模型，采用雅可比型扩散项，同时受预测和约束上界的驱动。我们严格证明了通过对漂移项中时变均值回归参数施加条件，该SDE模型强解的存在唯一性。通过阈值化处理归一化预测函数，保持均值回归参数的有界性。模型参数校准还通过采用新颖的迭代两阶段优化过程，对用户选择的似然函数近似形式进行优化，覆盖了归一化预测的阈值参数。另一创新贡献在于，通过结合控制变量法的定制核平滑技术，估计了无解析闭式解的预测误差过程转移密度。我们以乌拉圭2019年光伏（PV）太阳能日发电量及预测数据拟合模型，计算了日最大太阳能光伏发电量估计值。拟合了两种统计版本的约束SDE模型，分别以贝塔分布和截断正态分布作为转移密度的代理。实证结果包括通过间接推断方法生成的归一化太阳能光伏发电量模拟及其路径置信区间。通过应用创新的核密度估计技术对预测误差过程的转移函数进行分析，提供了两种选定统计近似对应的最优参数点的客观比较。